Quasi-bialgebra

Braided quasi-bialgebras

A braided quasi-bialgebra (also called a quasi-triangular quasi-bialgebra) is a quasi-bialgebra whose corresponding tensor category ${\displaystyle \mathcal{A}-Mod}$ is braided. Equivalently, by analogy with braided bialgebras, we can construct a notion of a universal R-matrix which controls the non-cocommutativity of a quasi-bialgebra. The definition is the same as in the braided bialgebra case except for additional complications in the formulas caused by adding in the associator.

Proposition: A quasi-bialgebra ${\displaystyle (\mathcal{A},\mathrm{\Delta },ϵ,\mathrm{\Phi },l,r)}$ is braided if it has a universal R-matrix, ie an invertible element ${\displaystyle R\in \mathcal{A}\otimes \mathcal{A}}$ such that the following 3 identities hold:

${\displaystyle ({\mathrm{\Delta }}^{op})(a)=R\mathrm{\Delta }(a)R^{-1}}$

${\displaystyle (id\otimes \mathrm{\Delta })(R)=({\mathrm{\Phi }}_{231}{)}^{-1}{R}_{13}{\mathrm{\Phi }}_{213}{R}_{12}({\mathrm{\Phi }}_{213}{)}^{-1}}$

${\displaystyle (\mathrm{\Delta }\otimes id)(R)=({\mathrm{\Phi }}_{321})R_{13}({\mathrm{\Phi }}_{213}{)}^{-1}{R}_{23}{\mathrm{\Phi }}_{123}}$

Where, for every ${\displaystyle a_1\otimes ...\otimes a_k\in {\mathcal{A}}^{\otimes k}}$, ${\displaystyle a_{i_1{i}_2...i_n}}$ is the monomial with ${\displaystyle a_j}$ in the ${\displaystyle i_j}$th spot, where any omitted numbers correspond to the identity in that spot. Finally we extend this by linearity to all of ${\displaystyle {\mathcal{A}}^{\otimes k}}$.

Again, similar to the braided bialgebra case, this universal R-matrix satisfies (a non-associative version of) the Yang–Baxter equation:

${\displaystyle R_{12}{\mathrm{\Phi }}_{321}{R}_{13}({\mathrm{\Phi }}_{132}{)}^{-1}{R}_{23}{\mathrm{\Phi }}_{123}={\mathrm{\Phi }}_{321}{R}_{23}({\mathrm{\Phi }}_{231}{)}^{-1}{R}_{13}{\mathrm{\Phi }}_{213}{R}_{12}}$

Twisting

Given a quasi-bialgebra, further quasi-bialgebras can be generated by twisting (from now on we will assume ${\displaystyle r=l=1}$) .

If ${\displaystyle {\mathcal{B}}_{\mathcal{A}}}$ is a quasi-bialgebra and ${\displaystyle F\in \mathcal{A}\otimes \mathcal{A}}$ is an invertible element such that ${\displaystyle (\epsilon \otimes id)F=(id\otimes \epsilon )F=1}$, set

${\displaystyle {\mathrm{\Delta }}^{\prime }(a)=F\mathrm{\Delta }(a)F^{-1},\mathrm{\forall }a\in \mathcal{A}}$

${\displaystyle {\mathrm{\Phi }}^{\prime }=(1\otimes F)((id\otimes \mathrm{\Delta })F)\mathrm{\Phi }((\mathrm{\Delta }\otimes id)F^{-1})(F^{-1}\otimes 1).}$

Then, the set ${\displaystyle (\mathcal{A},{\mathrm{\Delta }}^{\prime },\epsilon ,{\mathrm{\Phi }}^{\prime })}$ is also a quasi-bialgebra obtained by twisting ${\displaystyle {\mathcal{B}}_{\mathcal{A}}}$ by F, which is called a twist or gauge transformation. If ${\displaystyle (\mathcal{A},\mathrm{\Delta },\epsilon ,\mathrm{\Phi })}$ was a braided quasi-bialgebra with universal R-matrix ${\displaystyle R}$ , then so is ${\displaystyle (\mathcal{A},{\mathrm{\Delta }}^{\prime },\epsilon ,{\mathrm{\Phi }}^{\prime })}$ with universal R-matrix ${\displaystyle F_{21}R{F}^{-1}}$ (using the notation from the above section). However, the twist of a bialgebra is only in general a quasi-bialgebra. Twistings fulfill many expected properties. For example, twisting by ${\displaystyle F_1}$ and then ${\displaystyle F_2}$ is equivalent to twisting by ${\displaystyle F_2{F}_1}$, and twisting by ${\displaystyle F}$ then ${\displaystyle F^{-1}}$ recovers the original quasi-bialgebra.

Twistings have the important property that they induce categorical equivalences on the tensor category of modules:

Theorem: Let ${\displaystyle {\mathcal{B}}_{\mathcal{A}}}$, ${\displaystyle {\mathcal{B}}_{{\mathcal{A}}^{\prime }}}$ be quasi-bialgebras, let ${\displaystyle {\mathcal{B}}_{{\mathcal{A}}^{\prime }}^{\prime }}$ be the twisting of ${\displaystyle {\mathcal{B}}_{{\mathcal{A}}^{\prime }}}$ by ${\displaystyle F}$, and let there exist an isomorphism: ${\displaystyle \alpha :{\mathcal{B}}_{\mathcal{A}}\to {\mathcal{B}}_{{\mathcal{A}}^{\prime }}^{\prime }}$. Then the induced tensor functor ${\displaystyle ({\alpha }^{\ast },id,{\varphi }_2^F)}$ is a tensor category equivalence between ${\displaystyle {\mathcal{A}}^{\prime }-mod}$ and ${\displaystyle \mathcal{A}-mod}$. Where ${\displaystyle {\varphi }_2^F(v\otimes w)=F^{-1}(v\otimes w)}$. Moreover, if ${\displaystyle \alpha }$ is an isomorphism of braided quasi-bialgebras, then the above induced functor is a braided tensor category equivalence.

Usage

Quasi-bialgebras form the basis of the study of quasi-Hopf algebras and further to the study of Drinfeld twists and the representations in terms of F-matrices associated with finite-dimensional irreducible representations of quantum affine algebra. F-matrices can be used to factorize the corresponding R-matrix. This leads to applications in statistical mechanics, as quantum affine algebras, and their representations give rise to solutions of the Yang–Baxter equation, a solvability condition for various statistical models, allowing characteristics of the model to be deduced from its corresponding quantum affine algebra. The study of F-matrices has been applied to models such as the XXZ in the framework of the Algebraic Bethe ansatz.